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In Real Analysis by Folland,

the product σ-algebra $ \displaystyle \bigotimes_{\alpha \in A} \mathcal{M_{\alpha}} $ on $X$ is defined by the σ-algebra generated by $ \{ \pi ^{-1} _{\alpha} (E_{\alpha}): E_{\alpha} \in \mathcal{M_{\alpha}} \} $

where $ \pi _{\alpha} : \displaystyle \prod_{\alpha \in A} X_{\alpha} =: X \rightarrow X_{\alpha}$ is a coordinate map.

Prop 1.3: If $A$ is countable, then $ \displaystyle \bigotimes_{\alpha \in A} \mathcal{M_{\alpha}} $ is the σ-algebra generated by $ \{ \displaystyle \prod_{\alpha \in A} E_{\alpha} : E_{\alpha} \in \mathcal{M_\alpha} \} $

I want to find a counter example that if A is uncountable, then the above proposition doesn't hold.. but I'm totally stuck, please help! Thanks in advance.

Hana Kim
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  • If $A$ is uncountable and $X$ has more than one point then $x\in X\Rightarrow (x_{\alpha })_{\alpha \in A}\notin \mathscr B(X)^{A}$ – Matematleta Apr 01 '17 at 14:41
  • Can you tell me why it is so? i.e., $ (x_\alpha) {\alpha \in A} \in { \displaystyle \prod{\alpha \in A} E_\alpha : E_\alpha \in \mathcal{M_\alpha} }$, but $ (x_\alpha) {\alpha \in A} \notin \displaystyle \bigotimes{\alpha \in A} \mathcal{M_\alpha} $ ? – Hana Kim Apr 02 '17 at 00:06
  • I guess the idea is this: Singletons are closed in the product space, so they are in $\mathscr B (\prod X)$ but they are not contained in $\otimes\mathscr B(X)$ because every set in the latter is generated by countably many (measurable) rectangles in the product. – Matematleta Apr 02 '17 at 04:06

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